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To determine the equations for graphs 1 and 3, I'll analyze each graph's properties such as amplitude, period, and any phase shifts or vertical shifts. 

### Graph 1:
1. **Amplitude**: The graph reaches a maximum of \( y = 5 \) and a minimum of \( y = -5 \), so the amplitude is \( 5 \).
2. **Period**: The graph completes one cycle from \( -\pi \) to \( \pi \), giving a period of \( 2\pi \).
3. **Vertical Shift**: The graph oscillates around the \( x \)-axis (no vertical shift).
4. **Equation Form**: Since it’s a sine or cosine wave with no phase shift and standard period, the equation is:
   \[
   y = 5 \cos(x)
   \]
   Alternatively, it could also be represented as \( y = 5 \sin(x + \pi/2) \) due to the relationship between sine and cosine.

   So, the equation for **Graph 1** is:
   \[
   y = 5 \cos(x)
   \]

### Graph 3:
1. **Amplitude**: The graph reaches a maximum of about \( y = 3 \) and a minimum of about \( y = -3 \), so the amplitude is \( 3 \).
2. **Period**: The graph completes one cycle from \( -4\pi \) to \( 4\pi \), giving a period of \( 8\pi \).
3. **Vertical Shift**: The graph oscillates around the \( x \)-axis, so there is no vertical shift.
4. **Equation Form**: With the longer period of \( 8\pi \), the coefficient of \( x \) should be adjusted. A function with period \( 8\pi \) can be represented as \( \cos\left(\frac{x}{4}\right) \) or \( \sin\left(\frac{x}{4}\right) \).

   Therefore, the equation for **Graph 3** is:
   \[
   y = 3 \cos\left(\frac{x}{4}\right)
   \]

### Summary:
- **Graph 1**: \( y = 5 \cos(x) \)
- **Graph 3**: \( y = 3 \cos\left(\frac{x}{4}\right) \)

Would you like more details or have any further questions?

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### Related Questions:
1. What would the equation be if there was a vertical shift in Graph 1?
2. How would the equation change for Graph 1 if it had a phase shift?
3. How can we verify the period of a trigonometric function from its graph?
4. What if Graph 3 had a sine function instead of a cosine function?
5. How do we determine the period of trigonometric graphs with non-standard coefficients?

### Tip:
To find the period of a trigonometric function \( y = a \cos(bx) \) or \( y = a \sin(bx) \), use the formula \( \text{Period} = \frac{2\pi}{|b|} \).

Learn how to determine the equations for trigonometric graphs with given amplitude, period, and no vertical shift. This example focuses on identifying the cosine function for graphs with standard and extended periods. Suitable for students in Grades 10-12.

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